Generalization of the Superposition Principle

The superposition principle leads to the following generahzed relationship between the strain tensor and the stress tensor for viscoelastic systems  

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives 

It can easily be seen that the retransformation of this equation gives Eq. (5.73). In the same way, the application of the correspondence principle to Eq. (4.118) leads to the expression 

In other words, independently of the viscoelastic history in the linear region, the tensile compliance function can readily be obtained from both the shear and bulk compliance functions. For viscoelastic solids and liquids above the glass transition temperature, the following relationships hold when t oo J t) t/T[ [Eq. (5.16)], D t) = y Jt [Eq. (5.21)], and D t)J t)/ . These relations lead to r 3t that is, the elongational viscosity is three times the shear viscosity. It is noteworthy that the relatively high value of tensile viscosity facilitates film processing. 

Laplace transform of the tensile relaxation modulus can be obtained from Eqs. (5.62) and (5.80). The pertinent expression is 

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